Ultraproducts As a Tool in the Model Theory of Metric Structures

Ultraproducts as a tool in the model theory of metric structures

C. Ward Henson University of Illinois

These slides are posted at faculty.math.illinois.edu/~henson Preliminary comments

Ultraproducts play a more signiﬁcant role in model theory of metric structures than in classical model theory of discrete, algebraic structures. In the metric setting, ultraproducts do more than proving compactness. They provide an important experimental tool for studying metric structures model theoretically. Especially toward verifying axiomatizability of classes of structures and clarifying what is deﬁnable. Early uses of ultraproducts

hints in work of Skolem (1934) and von Neumann (1942). a metric ultraproduct occurs in work of Wright, a student of Kaplansky (1954; Ann. of Math.) for discrete structures, the work ofLo´swas published in 1955; Robinson’s nonstandard analysis in 1961. ultraproducts of some operator algebras in Sakkai (1962), McDuﬀ (1969), Connes (1974), ... . Banach space ultraproducts in Krivine (1967), nonstandard hulls in Luxemburg (1968); used extensively by Lindenstrauss, Pisier, Johnson, ... and studied by Stern, Heinrich, ... . metric geometry: (ultralimits) Gromov (1981), (asymptotic cones) van den Dries–Wilkie (1984). A crash course in continuous logic

Signatures L specify moduli of uniform continuity for their function symbols F and predicate symbols P. L-structures: complete metric spaces equipped with constants, functions, and predicates that interpret the symbols of L. predicates are [0, 1]-valued; predicates and functions must satisfy the given moduli. L-terms: built as usual from constants and function symbols.

Atomic L-formulas: of the form P(t1,..., tn) and d(t1, t2). L-formulas: obtained by closing under continuous connectives u : [0, 1]n → [0, 1] and quantiﬁers sup and inf. A theory T in L is a set of axioms, which are expressions of the form “σ = 0” where σ is an L-sentence (formula with no free variables). We identify T with the set of sentences σ. Ultraproducts deﬁned

(Mi | i ∈ I ) are L-structures, U an ultraﬁlter on I . Q We let N0 = i∈I Mi as a set; its members are (a) = (ai | i ∈ I ), ai ∈ Mi . We interpret the symbols on N0: N M F 0 (ai | i ∈ I ),... = F i (ai ,...) | i ∈ I ∈ N0 N M P 0 (ai | i ∈ I ),... = limi→U P i (ai ,...) ∈ [0, 1]

dN0 is only a pseudometric; E((a), (b)) :⇔ dN0 ((a), (b)) = 0 is an equivalence relation and the quotient N0/E has canonical metric and interpretations of the symbols of L.

We deﬁne the ultraproduct to be N0/E (which is complete) Q and denote it by N := i∈I Mi /U. The image of (a) ∈ N0 in N is denoted (a)U ; note that h i N0 (a)U = (b)U ⇔ 0 = lim d(ai , bi ) = d (a), (b) . i→U Properties of ultraproducts

Lo´s’sTheorem : for every L-formula ϕ(x, y,...) and elements Q (a)U , (b)U ,... ∈ N = Mi /U:

N Mi ϕ (a)U , (b)U ,... = lim ϕ (ai , bi ,...). i→U Corollary The Compactness Theorem for continuous logic of metric structures. Elementary

Let M, N be L-structures. Let A ⊆ M, B ⊆ N and f : A → B. Deﬁnition (a) M and N are elementarily equivalent if σM = σN for all L-sentences σ. We write M ≡ N. (b) The map f is elementary if (M, ha | a ∈ Ai) ≡ (N, hf (a) | a ∈ Ai).

Examples Any isomorphism from M onto N is elementary. The diagonal map DM : M → MU is elementary. Any restriction of an elementary map is elementary The composition of elementary maps is elementary. The inverse of an elementary map is elementary. Let M, N be L-structures. Let A ⊆ M, B ⊆ N and f : A → B. Keisler-Shelah Theorem M ≡ N if and only if there is an ∼ ultrafilter U such that MU = NU . Corollary The map f is elementary iff there is an ultrafilter U and an isomorphism J from MU onto NU such that J ◦ DM = DN ◦ f on A. Definition A class C of L-structures is axiomatizable if there is an L-theory T such that C is the class of all models of T ; that is, C = {M | M is an L-structure and σM = 0 for all σ ∈ T }.

Theorem For any class C of L-structures, the following are equivalent: (a) C is axiomatizable. (b) C is closed under isomorphisms, ultraproducts, and ultraroots.

M is an ultraroot of N if there is an ultraﬁlter U such that N is isomorphic to MU . T -formulas

Let T be an L-theory and x = x1,..., xn a tuple of distinct variables.

A T -formula ϕ(x) is a sequence (ϕn(x)) of L-formulas such M that ϕn (a) converges as n → ∞, uniformly for all models M of T and all a ∈ Mx . M The interpretation ϕ of a T -formula ϕ(x) = (ϕn(x)) in a model M of T is deﬁned by M M x ϕ (a) := limn ϕn (a) for all a ∈ M . We refer to the interpretations of T -formulas in the models of T (as above) as T -predicates.

Example: for any L-formulas (θn(x)), the weighted sum P −n M n 2 θn (x) is a T -predicate. Uniform asssignments of sets to models of a theory

Let T be an L-theory; ﬁx a tuple of distinct variables x = x1,..., xn. We will denote by X any operation that assigns a subset x X(M) ⊆ M to each model M of T . We refer to such an X as a uniform assignment of sets to models of T .

For example, if ϕ(x) is a T -formula, we consider Xϕ defined by x M Xϕ(M) := {a ∈ M | ϕ (a) = 0}. Such an operation will be referred to as a T -zeroset in the variables x. We note the following properties of T -zerosets X = Xϕ: (iso) If M, N are models of T and J is an isomorphism of M onto N, then J(X(M)) = X(N). (diag) If M is a model of T and U is an ultrafilter, with D : M → MU the diagonal embedding, then x D(X(M)) = D(M ) ∩ X(MU ). These two conditions yield that if M, N are models of T and J is any elementary embedding from M into N, then x J(X(M)) = J(M ) ∩ X(N). For model theorists: a uniform assignment X satisfies (iso) and (diag) if and only if there is a set X of x-types over the theory T such that for any model M of T we have x X(M) = {a ∈ M | tpM (a) ∈ X }. Properties of X that we discuss later correspond to properties of X in the topometric space Sx (T ). Characterization of T -zerosets

Let T be a theory in a countable signature L and let x = x1,..., xn be a tuple of distinct variables. Let X be a uniform assignment of sets to models of T . Theorem The following are equivalent: (1) There exists a T -formula ϕ(x) such that X = Xϕ. (2) X satisﬁes (iso) and (diag), and the containment Q Q X(Mi )/U⊆ X( Mi /U) Q holds for every ultraproduct Mi /U of models of T .

Note that the conditions in (2) can be veriﬁed by understanding how the assignment X behaves under isomorphisms and on ultraproducts of models of T ; this will not generally require a full understanding of the behavior of X(M) for arbitrary models of T . T -Deﬁnable sets

Let T be a theory in a countable signature L and let x = x1,..., xn be a tuple of distinct variables. Let X be a uniform assignment of sets to models of T . Definition We say X is T -definable if the class of T -predicates is closed under the quantifiers supx∈X(M) and infx∈X(M).

For a T -deﬁnable assignment X, the central idea is that we have a natural concept of induced model theoretic structure on X(M) that is obtained uniformly from L-formulas applied to M. Among other things, this can assist in providing an explicit axiomatization of the models of T . Characterization of T -deﬁnable sets

Let T be a theory in a countable signature L and let x = x1,..., xn be a tuple of distinct variables. Let X be a uniform assignment of sets to models of T . Theorem The following are equivalent: (1) X is T -definable. (2) X satisfies (iso), and the equality Q Q X(Mi )/U = X( Mi /U) Q holds for every ultraproduct Mi /U of models of T . M (3) The predicate P(x) defined by P (a) := dist(a, X(M)) for all models M of T and all a ∈ Mx is a T -predicate.

Note that condition (diag) is omitted in (2); it follows from the ultraproduct equality in (2), applied to ultrapowers. Let T be a theory in a countable signature L and let x = x1,..., xn be a tuple of distinct variables. Let X be a uniform assignment of sets to models of T . Corollary The following are equivalent: (1) X is T -deﬁnable. (2) X is a T -zeroset and some T -predicate P(x) with X as its zeroset has the almost-near property; namely, for all epsilon > 0 there exists δ > 0 such that for all models M of T and all a ∈ Mx , we have M P (a) < δ implies dist(a, X(M)) ≤ . (3) X is a T -zeroset and every T -predicate P(x) with X as its zeroset has the almost-near property. Example 1

Let C be the class of pointed metric spaces (M, c) of diameter ≤ 1 in which the metric d is an ultrametric (i.e., d satisﬁes d(x, z) ≤ max(d(x, y), d(y, z))). Let T be the theory of C.

Fix r in the interval 0 < r < 1 and for every (M, c) in C let X(M) = {a ∈ M | d(c, a) ≤ r}.

Obviously X is a T -zeroset. However, X is not T -definable. Proof: Let (M, c) be a member of C in which there exist elements (an) such that infn d(c, an) = r and r < d(c, an) for all n ∈ N. Let U be a nonprincipal ultrafilter on N. The element a = (an)U of the ultrapower (M, c)U satisfies d(c, a) = r. However, the ultrametric inequality prevents the existence of any sequence (bn) from M that satisfies a = (bn)U and d(c, bn) ≤ r for all n ∈ N. Example 2

Let L be a countable signature and T an L-theory. Recall that we say T is ω-categorical if T has a unique separable model, which is not compact. Proposition The following are equivalent: (1) T is ω-categorical. (2) Every T -zeroset is T -deﬁnable.

Proof. x For each model M0 of T and a0 ∈ M0 , let X be defined on models M of T by x X(M) := {a ∈ M | (M, a) ≡ (M0, a0)}. Obviously X is a T -zeroset. The omitting types theorem for continuous logic implies that X is T -definable iff X(M) 6= ∅ for all models M of T . The characterization of separable categoricity then implies the equivalence of (1) and (2). Example 3

This example comes from work in progress with Yves Raynaud, in which we are constructing new families of uncountably categorical Banach spaces. These spaces are asymptotically hilbertian in a strong sense. Deﬁnition A Banach space E is asymptotically 2-hilbertian if every ultrapower of E has the form EU = D(E) ⊕2 H for some Hilbert space H ⊆ EU , where D is the diagonal embedding of E into EU .

Here we only have time to discuss how the Hilbert part of EU can be identiﬁed. Let Tb denote a theory that axiomatizes the class of all unit balls of Banach spaces over R. For each such Banach space E, let B(E) denote the unit ball of E, considered as a model of Tb. For each a ∈ B(E) we let hai denote the linear subspace of E generated by a. We call hai a 2-summand of E if there is a subspace F of E such that E = F ⊕2 hai; note that if this equation holds, then F is uniquely determined by a. Note further that if E = F ⊕2 H and H is a Hilbert space, then hai is a 2-summand of E for every a ∈ H.

For each model B(E) of Tb, we deﬁne X(B(E)) := {a ∈ B(E) | hai is a 2-summand of E}.

Proposition

The set assignment X is a Tb-zeroset but it is not Tb-deﬁnable. To prove that X is a Tb-zeroset, we verify the conditions in the ultraproduct characterization of zerosets. Condition (iso) is obvious. Condition (diag) follows from the easy stronger fact that if a ∈ B(E1) and E1 ⊆ E2, and if E2 = F ⊕2 hai, then E1 = (F ∩ E1) ⊕2 hai. The ultraproduct containment condition holds because the ⊕2 direct sum operation commutes with the ultraproduct construction.

To prove that X is not Tb-deﬁnable, consider E = ⊕2(En | n ∈ N), where each En is polyhedral and 2-dimensional, with the unit sphere of En approximating the unit circle more and more closely as n → ∞. Then X(B(E)) = {0}, whereas E is asymptotically 2-hilbertian, so X(B(EU )) will be very large. Thus the ultraproduct equality condition fails to hold.